Expanding the horizon of molecular electronics via nanoparticle assemblies

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E XPANDING THE H ^ORIZON

VIA

M ^OLECULAR E ^LECTRONICS

OF

N ANOPARTICLE A ^SSEMBLIES

Laetitia Bernard

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!

Expanding the Horizon of Molecular Electronics

via

Nanoparticle Assemblies

A thesis submitted in accordance with the requirement of the University of Basel for the degree of

Doctor of Philosophy

in the Faculty of Philosophy and Natural Science by

Laetitia Bernard

from Montreux (Vaud), Switzerland

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Expanding the Horizon of Molecular Electronics via Nanoparticle Assemblies Laetitia Bernard

Supervisor :

Prof. Dr. Christian Sch¨onenberger Referees :

Prof. Dr. Marcel Mayor Prof. Dr. Bernard Doudin Dean :

Prof. Dr. Jacob Wirz

Basel, September 2006

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If you look for Nature’s secrets in only one direction, you are likely to miss the most important secrets, those you did not have enough imagination to predict.

Freeman Dyson

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Table of Abbreviations

ac Alternating Current

C12 Dodecanethiol (C12H25SH) C16 Hexadecanethiol (C16H33SH) C₈ Octanethiol (C₈H₁₇SH)

dc Direct Current

DEP Dielectrophoresis

DFT Density Functional Theory EBL Electron-Beam Lithography FFT Fast Fourier Transform FTIR Fourier Transform Infrared FWHM Full Width at Half Maximum HF Hartree-Fock calculation

HOMO Highest Occupied Molecular Orbital

ip In Plane

IR Infrared

I-V Current-voltage characteristic LT Low Temperature (4.2 K)

LUMO Lowest Unoccupied Molecular Orbital MCBJ Mechanically Controllable Break Junction MPC Monolayered Protected Cluster

oop Out Of Plane

OPE Oligo(Phenylene Ethynylene) OPV Oligo(Phenylene Vinylene) PDMS Polydimethylsiloxane PMMA Poly(Methyl Methacrylate)

RMS Root-Mean-Square

RT Room Temperature (20-25^◦C) SAM Self Assembled Monolayer SEM Scanning Electron Microscopy SPR Surface Plasmon Resonance STM Scanning Tunnelling Microscopy TEM Transmission Electron Microscopy THF Tetrahydrofuran (C4H8O)

TLC Thin Layer Chromatography

US Ultrasonic

UV Ultraviolet

UVL Ultraviolet Lithography

XPS X-ray Photoelectron Spectroscopy

ZBA Zero Bias Anomaly

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A few Common Symbols

β tunnelling Decay Coefficient ˚A⁻¹

c speed of light 3·10⁸m/s

C capacitance F

d inter-particle distance nm

∆ tunnelling distance m

δ deformation vibration resonance cm⁻¹

e electronic charge −1.6·10⁻¹⁹C

E energy J, eV

E~ electric field V/m

² energy level J, eV

ε permittivity dimensionless

Ea activation energy eV

Ec charging energy eV

E_F Fermi energy eV

f(E) Fermi-Dirac function dimensionless

f frequency Hz

f filling factor dimensionless

FDEP dielectrophoresis force N

φ tunnelling barrier energy eV

G =1/R conductance Ω⁻¹

G0 quantum of conductance Ω⁻¹

γ damping constant s⁻¹

Γ energy broadening eV

h Planck’s constant 6.63·10⁻³⁴Js

~ =h/2π

I current A

J current density A/m²

k =1/λwave number cm⁻¹

k_B Boltzmann constant 0.087 meV/K

l molecular length nm

λ wavelength nm

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viii Table of Contents

λF electron mean free path nm

m free electron mass 9.1·10⁻³¹kg

ν stretching vibration resonance cm⁻¹

ω frequency s⁻¹

ω_p plasmon frequency s⁻¹

~p dipole moment C·m

p(t) percolation probability dimensionless

P power W

R resistance Ω

σabs absorption cross section m⁻²

t time s

T temperature K

T transmission function dimensionless

τ percolation time min

V bias voltage V

Z impedance Ω

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1 A Clever Combination : Organic Molecules &

Nanoparticles

In view of current and future technological developments, Si-based mi- croelectronics faces a constant need for miniaturisation. However, its evolution towards nanoelectronics presents severe physical and economic limitations. Several potential alternatives to supplement or to replace the current technology have been investigated, like silicon MOSFETs^1,2,3, or techniques based on novel materials such as carbon nanotubes^4,5. An- other alternative is based on the use of single organic molecules, acting as electronic switches and storage elements⁶. Indeed, molecules exhibit inherent advantages over the current devices, making them appear as the ideal object for designing future high-density electronic devices. They are several orders of magnitude smaller than present feature sizes. They may be produced in large amounts absolutely identically and in a cheap way by chemical synthesis. Their physical properties are tunable by their structures. In addition, they have the potential to self-organise into regular 2D or 3D patterns. Therefore, complete systems for information processing may be built from basic functional units consisting of molecules acting as logic devices.

The use of molecules to perform electronic functions was first proposed in 1971 by Kuhn⁷, followed by Aviram and Ratner⁸ in 1974, who, based on theoretical assumptions, envisioned the use of a donor-acceptor molecule to produce a molecular rectifier. At this time, any realisation of a molecular device was technologically absolutely unfeasible. Nowadays, the development of this visionary concept and its extension into the experimental domain has become the broad research area calledMolecular Electronics⁹, as demonstrated by the increasing number of citations of this early paper (Figure1.1). The real beauty of this concept is the per- spective that specific electronic functions of a device may be adjusted by the design of the chemical structure of the molecules^10,11. Moreover, additional tunability by other stimuli, like voltage, light or magnetic field

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2 A Clever Combination : Molecules &Nanoparticles

can be envisioned.

In this research area, two very different approaches are under exten- sive investigation: the bulk molecular systems and the single molecular systems. The first one is defined by characteristic dimensions which are much larger than the sizes of the molecules. Consequently, most of the molecules are in contact with other molecules, making them not individ- ually addressable. The properties of this ensemble must be considered as a whole. In contrast, single molecular systems aim for individual contact to single molecules or small arrays of perfectly ordered molecules. This second approach strongly differs from the first one, as it tries to utilise the physical properties of single molecules for nanosized electronic devices.

Number of citations

1975 1980 1985 1990 1995 2000

0 20 40 60 80 100 120

Year A. Aviram & M.A. Ratner,

Chemical Physics Letters 29, 277 (1974)

Figure 1.1: Time evolution of the number of citations of Aviram and Rat- ner’s paper from 1974, precursor of the new fieldMolecular Electronics.

A basic requirement for Molecular Electronics is the connection of the molecules to the outside world. In the case ofbulk molecular systems, this is achievable thanks to conventional UV or electron-beam lithography, which enables the fabrication of micrometre-scale contact patterns (down to 20 nm). For single molecular systems, an electrode pair with nanometre-sized spacing to contact a single molecule is needed. Although the development of such a device is challenging, the technological advance would provide the possibility to address one or few molecules. Several techniques were investigated, such as STM^12,13,14, mechanically controllable break junctions (MCBJ)^15,16, cross-bar arrays^17,18,19, electromigra- tion^20,21 or mercury droplet²². Quantum phenomena such as Coulomb blockade²⁰, Kondo effect²¹ and negative differential conductance¹⁰ have already been observed. However, reproducibility and stability of such re-

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3 sults still remain uncertain²³. Indeed, the mobility of electrodes’ atoms makes the metal-molecule junction unstable, strongly affecting the properties of the device, meaning that large ensembles of molecules would therefore be more suitable for technological applications. Devices devel- oped using this second approach are already present on the market, such as for example liquid crystal displays or organic light emitting diodes.

Although tremendous progress was achieved on both approaches over the past few decades^24,25,26, the understanding of current transport through molecules remains uncomplete. Both experimental and theoretical evidence show that the electronic properties of molecular junctions are not only dictated by the molecules contacted, but also depend on the anchor- ing groups and on the electrodes forming the junction^23,27. Thus, along with the properties of the molecule itself, those of the block “contact- molecule-contact” also have to be considered. Introducing such building blocks into functional electronic circuits remains a demanding task, re- quiring innovative approaches in fabrication philosophy and circuit structure^17,28. Nanometre-size metallic nanoparticles as intermediate electrodes can be used as an elegant solution to build molecular junctions with well-defined geometry and electronic properties. Indeed, combining metallic nanoparticles with molecules to form such building blocks would overcome the size mismatch between nanometre-scale molecules and micrometre-scale electrodes29,30,31,32,33,34,35.

We propose in this PhD Thesis a multidisciplinary study, interlink- ing chemistry (synthesis of nano-objects constituting building blocks), engineering (design of circuits based on the assembly of such building blocks) and physics (understanding of the ensemble properties). Con- cretely, two projects based on the assembly of nanoparticles were performed, on one hand, one-dimensional assembly, and on the other hand, two-dimensional assembly. The first one combines nanolithography and electrostatic trapping of colloids to achieve the fabrication of conducting nanoparticle chains. The second project consists of the self-assembly and the micro-contact printing of ligand-protected nanoparticles, followed by in-situ ligand exchange reactions. This fabrication method enables the preparation of stable two-dimensional networks of molecular junctions.

Remarkably, this approach, combining thesingle molecular approachand the bulk molecular approach introduced above, takes the advantages of addressability of the first one and stability of the second one.

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4 A Clever Combination : Molecules &Nanoparticles

Outline

This dissertation is organised as follows: a concise introduction to molecular electronics and to nanoparticle physics is given in Chapter2. The 1D approach is presented in the Part 1. Whereas the concept of fabrication, based on theactrapping technique, is described in Chapter3, the electrical characterisation and the technical limitations are discussed in Chap- ter4. The Part 2 presents the 2D approach. The fabrication philosophy - the self-assembly, as well as the circuit design - the micro-contact printing followed by shadow evaporation, are introduced in Chapter 5. Electrical and optical characterisations follow in Chapter6 and Chapter 7, respectively. Discussion on the advantages and limitations of such devices end the chapter. The dissertation is closed by Chapter8with general remarks and prospective ideas for this project. For detailed pieces of information, technical data on the different fabrications and measurements and com- plementary experiments, we report the reader to the Appendices.

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2 A Short Guide to Organic Molecules & Nanoparticles

2.1 Insights on Molecular Physics

2.1.1 What is the Resistance of a Molecule ?

The first question one comes to when considering the field of Molecular Electronics, is : “What is the resistance of a molecule ?”. The term of resistance was used to define, in bulk material, a macroscopic property related to a scattering process of the electrons flowing through the material, namely a kind of friction effect. Following the top-down approach, scientific interest has moved to smaller and smaller objects, nonetheless keeping the same concepts established for the bulk material. At a certain point, this gave rise to an interesting phenomenon, calledballistic trans- port, where electron transport was free of scattering. Thus, this appears confusing, since the so-called resistance is not clearly defined in such a situation. Once the existence of a finite number of discreet channels of conduction in small objects demonstrated^36,37, Landauer³⁸ has been the one to find a way out of this. He quantum mechanically showed that the conductance could be described in term of the sum of the transmission functions T_i for all the channelsi:

G= 2e² h

XN

i=1

T_i (2.1)

withG₀= ^2e_h² the quantum conductance, namely the highest possible conductance for one electron transport channel^∗.

This formalism is nowadays widely used in the field ofMolecular Elec- tronics, as gives evidence the increasing number of review articles report- ing it^39,40. In the case of a molecule coupled to metal electrodes, two

∗Note that the factor 2 in this expression accounts for the two spins.

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6 A Short Guide to Organic Molecules &Nanoparticles

aspects appear to be crucial for the transmission. The first aspect concerns the molecule itself. It was suggested in the introduction that one can choose the electrical properties of a molecular junction by synthesising a specific molecule, having the desired properties. There is a wide range of possibilities for molecules : they can be more insulating, more conducting, or have some functionality. The second aspect is the nature of the contact between the molecule and the electrodes. The metal-molecule- metal coupling is at least of equal importance as the molecular structure itself. The control of this coupling is a major issue. Indeed, it has to be strong enough to allow the current flow, but weak enough the pre- serve the intrinsic electronic characteristics of the molecule. These two aspects, the processes involved and the limit of their understanding are briefly described in this section.

2.1.2 The Organic Molecule

The simplest molecules from the electronic point of view are molecular wires: a molecular scale extension of macroscopic metal wire. In any organic molecule, however, electron transport is expected to take place through the frontier orbitals of the molecule, being the closest to the Fermi levels of the electrodes. The molecule size, as well as the presence ofπ- systems, decrease the energy between the frontier orbitals, hence making the molecule a better conductor. Therefore, non-conjugated molecules (neitherπ-system norσ-system) like alkanes are good insulators and conjugated molecules containing for example large delocalisedπ-systems like polyenes allow the transport of electric charges (see Figure2.1).

n n

A B

Figure 2.1: Schematics of two molecular wires. A. Alkane (not conjugated molecule); B. Polyene (conjugated molecule). Although their structure is very similar, the first one is insulating, while the transport of electrons is possible through the second one.

Both the insulating and conducting properties are of importance, and are used in different applications. However, those are not the only properties which can be optimised. The rigidity of the structure is of similar importance, for example to prevent short circuiting of separated units through space. Rigid non-conjugated molecules are rather seldom.

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2.1Insights on Molecular Physics 7

Alkanes, known for their insulating properties, lack the required rigidity.

Cyclic π-systems meet the rigidity conditions but are good conductors.

However, the delocalisation of the π-systems depends strongly on the torsion angles between the subunits (aromatic rings for example)⁴¹. Two neighbouring subunits with perpendicular π-systems reduce their elec- tronic communication (see Figure 2.2) by reducing the overlap of the electronic orbitals. Hence, the conduction of electrons through such a molecule is reduced and the molecule appears less conductive.

Figure 2.2: Aromatic rings perpendicular to each other decrease signifi- cantly the overlap of theirπ-systems, hence, the overall conductance of the molecular unit.

Furthermore, the electron conduction through an aromatic ring also depends on the relative position of the linkage (see Figure2.3)⁴². While ortho- and para- connections are conjugated linkers, hence allowing a good conduction, themeta-position is not, as two consecutive single bonds disable the overlapping of theπ-orbitals. Therefore, the conduction is in this configuration decreased.

ortho-

meta- para-

Figure 2.3: On an aromatic ring, three relative positions of the linkage are possible. They are known as ortho-,meta- andpara- positions.

2.1.3 Molecular Electronics : The Contacts

To study and understand the electronic properties of a molecule, one first has to be able to electrically address it, hence to contact it (Figure2.4).

This is where the second aspect, the nature of the contacts, plays a crucial role. The usual way to electrically contact a molecule or an ensemble of molecules is to bind it to metallic electrodes. To ensure a mechanically

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strong bond between the electrodes and the molecule, binding groups acting as “alligator-clips” (generally sulfur atoms) are attached at both ends of the molecule, providing a covalent bond. However, let us note that although they are good mechanical bonds, they are not electrically transparent^43,44. The electrodes constitute the two leads, with Fermi energyE_F, to and from which the electrons may flow.

A V

Figure 2.4: How to contact and measure the electronic properties of an organic molecule ?

To describe the electron transfer mechanism through the molecule, one has to consider the energy levels diagram of the metal-molecule-metal system, as presented in Figure 2.5A. Whereas the metal leads present a continuum of energy levels, the molecule has discreet levels, also called molecular orbitals. In particular, one can distinguish a set of filled levels (underE_F) and a set of empty levels (aboveE_F). The electrons can flow if one level is nearby the Fermi level. This is in general not the case, since the lowest unoccupied molecular orbital (LUMO) is separated from the highest occupied molecular orbital (HOMO) by a gap of typically a few eV. A possibility to get electron transfer is to change the energy levels of the leads with respect to the molecular levels, by applying a bias voltage V in between them, as shown in Figure 2.5B. Hence, by moving apart from the Fermi level, the leads’ levels open an energy window of e·V, where the transfer of electrons is possible. Whereas the left contact tends to fill up all available states below E_F + ^eV/2, the right contact keeps emptying all states above E_F −^eV/₂, hence yielding in an electron flow through the molecule, from left to right. As long as levels are in the energy window, they and only them, allow the transfer of electrons. This flow might be more or less large, depending on the quality of the contacts.

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metallic lead

metallic lead EF

molecular orbitals

HOMO LUMO

A

metallic

lead metallic

lead EF

molecular orbitals

eV

B

V > 0

e- e-

Γ¹ Γ²

Figure 2.5: Energy diagram of a molecule in between two metallic leads with Fermi energy E_F. A. No voltage is applied at the edges of the junction. No electron can flow as no level lies nearby the Fermi energy.

B. By applying a voltageV, the leads’ energies shift with respect to the Fermi energy while the molecule orbitals remain at the same position.

An energy window of width eV opens. As soon as the first molecular orbital is included in this window, electrons are transferred from the left lead to the molecule and from the molecule to the right lead (electrons flow).

To quantitatively characterise the current one has to consider the typ- ical time needed for the electrons to flow from one lead to the molecule, and from the molecule to the second lead. This would be the objective quantification of the quality of the contacts. Hence, one may introduce the escape rates, or tunnelling rates, Γ₁/~ and Γ₂/~, corresponding to those characteristic times and representative of the coupling between the leads and the molecule, for the left and the right contacts respectively.

Physically, Γ_i/~ is the number of times per second that an electron suc- ceeds in flowing through the barrieri= 1,2. The tunnelling rates induce also a broadening of the molecular orbitals. Indeed, in the case of strong coupling, the molecular orbitals and the leads’ electronic states do overlap.

This yields a hybridisation of the electronic states, and thus a common delocalised electronic wave function extending over the whole junction.

This provokes a broadening of the energy levels of Γ = Γ₁+ Γ₂, as shown in Figure 2.6, as well as a shift of the molecular orbitals. Hence, one has to consider rather the molecular density of states :

D_²(E) = 1 2π

Γ

(E−²)²+ (Γ/2)² (2.2)

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instead of the discreet molecular levels, with²the levels’ energy. The expression of the current flowing across the junction is then the integral over the whole energy range, and is expressed in the case of symmetric barriers by :

I(V) = e

~· Z∞

−∞

D_²(E)·Γ₁Γ₂ Γ ·

n

f(E+eV

2 )−f(E−eV 2 )

o

dE (2.3)

withf(E) theFermi-Dirac function. Note that because of the broadening, the molecule might exhibit a higher conductance than what one could expect by considering non-degenerated levels, as one can see in Equation (2.3). IdentifyingD_²(E)·^Γ¹_Γ^Γ² = ¹_πT(E), (2.3) is equivalent to the energy dependent Landauer formula, which reads :

I(V) = 2e h ·

Z∞

−∞

T(E)· n

f(E+eV

2 )−f(E−eV 2 )

o

dE (2.4)

metallic

lead EF

molecular density of states

eV

B

V > 0

metallic

lead EF

molecular orbitals

eV

A

V > 0 Γ¹

Γ²

Γ¹

Γ² Γ

Figure 2.6: Transport through a metal-molecule-metal junction. A. The molecule is decoupled from the leads and the levels are sharp. B. The coupling to the leads is good, hence, a broadening of the levels occurs, as well as a shift of the level, with respect toEF. The coupling to the leads is represented by Γ1and Γ2for the left and right leads respectively.

Let us remark that the current is limited to a maximum value. Indeed, this maximum is reached when the transmission function is maximum T(E) = 1. Hence, the integral (2.4) resumes, at T = 0 K, to :

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I_max(V) = 2e h ·

1 2eV

Z

−¹₂eV

1dE= 2e²

h V (2.5)

defining thus the quantum of conductance presented in Equation (2.1) for one channel, possibly expressed in term of resistance :

R₀ = 1 G₀ = h

2e² ≈12.9kΩ (2.6)

This resistance is a characteristic constant in the field of Molecular Electronics^∗.

Finally, note that in the case of a non-conducting molecule, i.e. whose HOMO-LUMO gap is so large that no states can be found in the energy window, one expect to have either no conduction at all, or tunnelling, directly from one lead to the other. The tunnelling current decays ex- ponentially with the distance d, and is described by the relation (2.7) at low-bias, Φ being the energy barrier andm the electron mass.

I ∝ 1

de⁻^~²^√^2mΦ·d (2.7)

For convenience, one may introduce the tunnelling decay coefficient β = ²_~√

2mΦ. As a matter of fact, it was shown that even though a molecule is in principle non-conducting, the tunnelling across it is different than the tunnelling through vacuum. Hence, the molecule, just by the presence of its electronic density, influences the transfer of electrons. This contribution appears in the current expression (2.7) through the energy barrier Φ, depending on the molecule and its levels’ position with respect to the leads E_F⁴⁰. This effect is known in the chemistry and physics communities as themediated tunnelling, orsuper-exchange.

Commonly and in the case of this thesis, gold is chosen for the leads, because of its noble metal properties (stable, non-reactive). However, other noble metals can be used, like palladium or platinum^45,46, and might present some advantages, such as lower mobility or enhanced wet- ting properties. Sulfur is widely used as binding chemical group, since it provides a strong covalent bond with gold, ensuring the mechanical stability of the contact, and a fast charge transfer⁴⁷. However, different binding groups are currently under investigation, like the dithiocarba-

∗Note that the resistance of one single atom-atom junction is typically of orderR0.

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mate (−N CS₂)⁴⁴, the cyanide (−CN)^48,49, or the selenium (−Se)⁴³, which was recently shown to allow a better coupling of the molecule with the electrodes. Some theoretical analysis of metal-molecule contacts make comparisons of different metals and different binding groups^50,51. This type of analysis discuss in particular the advantages and disadvantages of different “metal/binding group” pairs.

As a conclusion, both the electrical properties of the molecule and the nature of the molecular contacts are of major importance in the transport properties of the whole junction⁵². In particular, the resistance might vary strongly, if the molecule-metal bond is arranged differently⁵³. Consequently, the resistance, defined as the bias voltageV divided by the current I, concerns the full metal-molecule-metal junction, rather than the molecule alone. Further information on the electron transport in molecular wires can be found in various review papers^27,39,54.

2.1.4 Molecular Optics

The interaction of light with organic molecules leads to electronic, vibrational and/or rotational excitations, depending on the incoming photons’

energy. Under ultraviolet or even visible light, the energy transferred to the molecules is sufficient to excite their electrons. At lower energies however, typically for light in the infrared region, this is no more the case. Instead, the interatomic bonds can be excited, exhibiting stretching and deformation vibrations. Rotations of the molecules can be observed at even lower energies. Those mechanisms are briefly presented in this section, as well as how their study enables to access properties of the considered molecules.

UV-Visible Light Interaction with Molecules

Ultraviolet (200-400 nm) as well as visible light (400-800 nm) is able, via its absorption, to promote the excitation of molecular electrons between the energy levels corresponding to the molecular orbitals of the systems.

In particular, transitions involving π-orbitals, σ-orbitals and lone pairs (n ≡non-bonding) are of great importance^∗. The lowest possible energy transition is the one between the HOMO and the LUMO in the ground state. The more conjugated the system, the smaller the HOMO-LUMO gap, and therefore, the larger the wavelength. Hence, the study of the

∗Note that other transitions are possible, like transitions involving charge-transfer electrons, or involving d and f electrons.

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resulting absorption spectra, via UV-Visible spectroscopy, can lead typically to the identification of conjugated systems and the determination of their HOMO-LUMO gap.

σ (bonding) σ∗ (anti-bonding)

π (bonding) π∗ (anti-bonding) n (non-bonding)

Energy

Figure 2.7: Diagram of the molecular orbitals : electronic excitations that can occur in organic molecules in interaction with UV-visible light.

Figure 2.7 presents a diagram of the various kinds of electronic excitation that may occur in organic molecules. Out of the six transitions outlined, only the two lowest energy ones (the n-π and π-π transitions) can be achieved by the energies available in the 200 to 800 nm spectrum. These transitions need an unsaturated group in the molecule to provide the π-electrons. The σ-σ transitions show an absorbance maximum at 125 nm, whereas the n-σ transitions can be initiated in the range 150−250 nm, but the number of organic functional groups exhibiting the latter transitions is small.

Moreover, absorption of ultraviolet and visible light in organic molecules is restricted to certain functional groups, calledchromophores, that contain valence electrons of low excitation energy. The spectrum of a molecule containing these chromophores is complex. However, because of the superposition of rotational and vibrational transitions on the electronic transitions (see next section), a combination of overlapping lines makes the absorption appear as a continuous absorption band.

Infrared Light Interaction with Molecules

Infrared (IR) electromagnetic waves, whose frequencies are situated between 10¹⁰Hz and 10¹⁴Hz^∗, typically are in the range of energies for exciting the atomic bonds’ vibrations of molecules, as well as their rotation. Indeed, the interaction IR light-molecule may induce vibrations and rotations at specific energy levels. The set of the vibrational and

∗This corresponds to wavelengths from 1µm to 1 cm.

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rotational levels that a molecule can have is very specific firstly to its composition (which atoms do compose the molecule) and secondly to its configuration (how these atoms are linked together). Hence, the position and intensity of absorption bands in the IR of a substance are extremely specific to that substance and constitute its fingerprint. Hence, IR spec- troscopy is a powerful tool for molecular identification and can be used for the detection of specific molecules within a heterogeneous substance by spectral comparison^55,56,57. Another way to identify a specific molecule within a substance is to simulate the position and intensity of the absorption peaks, according to the molecular composition and shape. Electronic structure methods, like ab initio Hartree-Fock method (HF) or density functional theory (DFT), are increasingly used for modelling molecular properties that includes equilibrium structures, vibrational frequencies and intensities. Those methods give thus a good estimation of the infrared absorption of small molecules.

Here, to get a feeling on the IR light-molecule interaction, we present a simple quantum mechanics derivation for a diatomic molecule. It gives already a good indication on the possible levels and transitions for vibrations and rotations of the molecule.

A B

r

m S m

1 2

V(r)

r

0

Ed

r₀

Figure 2.8: A. Schematic of a diatomic molecule experiencing stretching vibrations. B. Interaction potential between two bonded atoms.

The potential expressing the interaction between two atoms is presented in Figure2.8B, with its minimum (equilibrium between attraction and repulsion forces) centred at r₀. For describing molecular vibrations of a diatomic molecule, one starts out from the simple harmonic oscillator model. The atoms are assumed to be points of mass m₁ and m₂, kept together by a weightless spring (see Figure2.8A) of initial lengthr₀ (without vibration). Let’s point out that the harmonic oscillator model supposes the potential to be parabolic and so, is valid only for small oscillations aroundr₀. Hence, considering the potentialV(r) = ¹₂mω(r−r_e)²,

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one may solve the Schr¨odinger equation with the hamiltonian being the sum of kinetic and potential energy :

h

− ~² 2m

d²

dr² +V(r) i

ψ_v(r) =E_vψ(r) (2.8) m= _m^m₁¹_+m^m²₂ is the reduced mass of the system,ω the pulsation of the incoming wave, ψ_v(r) are the eigen states of the system, corresponding the the eigen valuesE_v, the energy levels of vibrations. By resolving this equation, one finds the expression for these energy levels of vibrations :

E_v =

³ v+1

2

´

~ω (2.9)

with v = 0,1,2, . . ., the vibrational quantum number. This results shows first that the ground state, given by v = 0, is not zero. Second, one sees that the levels for vibration are quantised. Hence, only specific energies of the incoming wave, corresponding to the energy difference between two states |Ψ_vi and |Ψ_v⁰i, can make the molecule vibrate. Fur- thermore, only certain transitions are allowed, according to the selection rule : ∆v=v−v⁰ =±1⁵⁸. As the IR spectrum of a substance is usually plotted as a function of the wave number k= 1/λ, it is more convenient to express the vibrational levels in unit of wave vector instead of energy, by introducing a function G(v) given in (2.10).

G(v) = E_v hc =k

³ v+ 1

2

´

(2.10) Then, because of the selection rule, the only possible transitions are given by G(v)−G(v±1) =k, as represented in Figure 2.9A.

However, for larger oscillations, as the potential is not parabolic (see Figure2.8B), one has to consider rather the anharmonic oscillator model to estimate the vibrational levels of the diatomic molecule. It is equivalent to consider the harmonic oscillator added by a third-order pertur- bation⁵⁸. This leads to two differences : first the energy levels E_v get a third-order correction (subtracted) which increases with increasev, lead- ing to lesser separation between the levels for largerv. More importantly, the selection rule gets weaker and becomes ∆v=±1,2,3, . . ., due to eigen states mixing. It means that transitions between states that are not consecutive may also happen (see Figure 2.9B). However, those transitions, called overtones, have a much smaller probability to happen than the fundamental vibration, given by ∆v=±1.

For describing molecular rotations of a diatomic molecule, one uses

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G(v)

1/2 k

v = 1 v = 0 v = 2 v = 3 v = 4

A B ^G(v)

v = 1 v = 0 v = 2 v = 3 v = 4

3/2 k 5/2 k 7/2 k 9/2 k

Figure 2.9: A. Vibrational energy levels of the diatomic molecule with a harmonic oscillator model. B. Correction of the vibrational energy levels of the diatomic molecule with an anharmonic oscillator model.

z

y

x

m

2

1

θ

r

φ

S

Figure 2.10: Schematic of a diatomic molecule experiencing rotations.

the rigid rotator model. One considers the moment of inertia I = mr_e² of the molecule, hypothesising that the two atoms’ distance is fixed (see Figure 2.10). The hamiltonian,H=−^~_2I²[_∂θ^∂²2 +_tgθ¹ _∂θ^∂ +_sin¹2θ ∂²

∂φ²], in this case contains only kinetic energy and the Schr¨odinger equation has to be solved in spherical coordinates. The rotational energy levels found by solving this equation are :

E_j = j(j+ 1)~²

2I (2.11)

with j = 0,1,2, . . ., the rotational quantum number. This shows that also the rotational levels are quantised. However, the ground state

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for rotation, given by j = 0, is zero. The rotation also has to obey a selection rule : ∆j =±1⁵⁸. In a similar way as for vibration levels, one can express the rotational levels in unit of wave vector, by introducing a functionF(v) given by :

F(j) = E_j

hc =Bj(j+ 1) (2.12)

withB = _4πcI^~ defined as therotational constant. Then, because of the selection rule, the only possible transitions are given byF(j)−F(j±1) = 2B(j+ 1), as represented in Figure 2.11A. For rotation thus, the levels get a larger separation for largerj.

F(j)

2B 6B 12B 20B

j = 1 j = 0 j = 2 j = 3 j = 4

A B ^F(j)

j = 1 j = 0 j = 2 j = 3 j = 4

Figure 2.11: A. Rotational energy levels of the diatomic molecule with the kinetic moment approach. B. Correction of the rotational energy levels due to the centrifugal force.

In reality, the distance between the two atoms is not constant and increases with increasing rotation energy due to centrifugal force. This leads to a larger I and thus, a reduced ∆F. That is the reason why the level spacing does not increase as much as predicted by the rigid rotator model (see Figure2.11B).

Finally, as the molecule vibrates and rotates simultaneously, the global band diagram (see Figure 2.12) combines both and the transitions from one state|Ψ_v,ji to another|Ψ_v⁰_,j⁰i are given by :





k= ∆G+ ∆F =G(v⁰)−G(v) +B_v⁰j⁰(j⁰+ 1)−B_vj(j+ 1)

∆v=±1,2,3, . . .

∆j=±1

(2.13)

Remark that because of the coupling rotation-vibration, B in (2.13) depends on v : B_v = B −a(v+ 1/2), with a a constant. This expres-

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G(v) + F(j)

v = 1

v = 0 v = 2 v = 3 v = 4

2 3 4 j

Figure 2.12: Superposition of the vibrational (v) and rotational (j) energy levels of a diatomic molecule.

sion plus the two selection rules give the positions (wave number) of the allowed transitions, thus the positions of the absorption peaks in the IR spectra.

2.2 Insights in Cluster Physics

Clusters constitute an intermediate state of matter, composed by a few atoms up to millions of atoms, between the single atom on the one hand and the bulk matter on the other. Their diameter ranges typically from 1 nm to several tens of nanometres, and their packing can be different from the bulk matter. In this intermediate case, the range of interac- tions is in general larger than the size of the particles. Furthermore, they are characterised by a large surface/volume ratio. Those effects strongly influence their atomic and electronic structure, and this influence is of- ten dependent on the size of the cluster. Two different kinds of cluster- size effects can be distinguished : the intrinsic effects, due to specific changes in volume and surface material, and the extrinsic effects, which are collective size-dependent response to external environment, such as the collective electronic and lattice excitations. For large clusters (typically Ø≥10nm), the extrinsic effects play a dominant role, whereas for small clusters, the intrinsic effects become important.

Hence, the electrical and optical properties of clusters may strongly differ from the bulk properties but also from the single atom properties.

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2.2Insights in Cluster Physics 19

Their description cannot be made by pure solid state physics nor pure quantum physics. Specific theories and models have to be considered for such systems, taking into account the classical thermodynamics and electrodynamics and including the size and environmental effects.

As the clusters used in the present experiments are considered as

“large” clusters (≥10nm), and to keep this chapter concise, we focus on this particular case throughout this basic introduction. It is important however to keep in mind that electrical and optical properties change drastically when considering so called “small” clusters. For similar rea- sons, we restrict our description to metallic spherical clusters.

2.2.1 Cluster Electronics

Like for molecules, the electronic properties of small clusters (typically

≤3 nm) are described by discreet quantum energy levels (Figure2.13C).

This type of structures is described by the jellium model, where the ion cores are seen as a static positively charged skeleton and the conduction electrons as a homogeneous negative cloud. For larger cluster sizes, these bands tend to group in bunches of energy levels. The electronic properties are then given by a new type of band structure, each band consisting of a bunch of single levels (see Figure 2.13B). For even larger clusters, like those considered in this thesis, the energy spectrum becomes similar to those of bulk gold^∗ (Figure2.13A).

metal/colloid cluster/nanocrystal E

molecule E

E

A B C

Figure 2.13: Energy diagrams for bulk, intermediate and molecular systems : continuous, bunches of single levels and discreet levels, respectively.

∗A rough estimation gives in our case, namely with clusters of Ø 10−20 nm, levels spacing of∼10⁻⁴eV.

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However, such large clusters cannot be fully considered as bulk material⁵⁹. Besides the question of the energy spectrum, there are also some important size effects. For example, when the cluster size is of the order of the electron mean free path (typically λ_Au ≈3 nm at room temperature⁶⁰), scattering of electrons on the boundaries become an important contribution. Another effect is the inefficiency of electron screening, due to a low number of atoms, as compared to bulk material. This results in an increase of the ionisation potential, which is dependent on the size.

Number of reviews report on the relationship between particle size and electronic behaviours⁶¹.

2.2.2 Cluster Optics

The interaction of photons with clusters leads to electronic and/or vibrational excitations. The dominant optical features of metal clusters are those associated with the largest oscillator strengths, issuing from the collective electron excitations in the clusters. Those collective oscillations, also calledsurface plasmons, are at the origin of the dipole modes in the clusters, related to the dipole resonances in atoms. Although all electrons are oscillating with respect to the positive ion background, the main effect is the oscillating surface polarisation of the clusters. The optical features resulting from this effect consist of an absorption band centred around the surface plasmon resonanceω_sp, positioned in the visible range wavelengths. The resonant width and peak intensity depend, on the electron relaxation rates, modified from bulk values by scattering off the cluster surface, on the internal structure (grain boundaries in polycrystalline particles), and on interface effects. The surface plasmon resonance frequency ω_sp is closely related to the bulk plasma resonance frequencyω_p = (_ε^ne₀_m²_e)^−1/2, and the exact expression depends on the sam- ple geometry. For spherical clusters in vacuum⁶²,ω_sp=ω_p/√

3. However in general, this expression and more globally the interaction of light with clusters is of high complexity, since it strongly depends on the material, the surrounding medium, the size and shape of the clusters. The first electrodynamics general solution was calculated byGustav Mie in 1908⁶³. This scattering theory (also called Mie Theory) is still extensively used nowadays to model cluster-light interaction (c.f. Section 7.1).

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2.2Insights in Cluster Physics 21

Mie Theory

The solution proposed by Mie divides the problem into two parts : an electromagnetic part, treating the general interaction between light and a metallic cluster, and a material part, circumvented by introducing a phenomenological dielectric function ε(ω, r) describing the “nature” of the cluster.

For the derivation of the general solution of the light diffraction of a single metallic sphere, Mie applied Maxwell’s equations with appropriate boundary conditions in spherical coordinates using multipole expansions of the incoming electric and magnetic fields. The solution was based upon the determination of scalar electromagnetic potentials Π, solving the wave equation ∆Π +|~k|²Π = 0, and whose solution in spherical coordinates can be expressed as Π(r, θ, φ) = R(r)Θ(θ)Φ(φ). These three functions were found to be a Bessel function, a Legendre polynomial function and a sinusoidal function for the r, θ, and φ dependencies respectively. These potentials constitute a basis of vector harmonics, allowing the expansion of an electromagnetic plane wave in such a spherical basis E~ = P^∞

m=0

P∞

n=mA_mnΠ_mn. Here, nrepresents the order of the expansion (pole) and m is the relative refractive index given by m = _N^N_m^c,N_c and N_m being the index of refraction of the cluster and the medium respectively. The so called scattering coefficients A_mn are determined by applying the boundary conditions, solved by considering the internal and scattered fields. The full derivation can be found elsewhere⁶⁴. The two sets of scattering coefficients, corresponding to the electric modes (a_n) and the magnetic modes (b_n), are given by the Bessel functions of first and second kind j_n and h_n=j_n±iy_nconstituting the R(r) function:





a_n= ^m²^jⁿ^(mx)[xjⁿ^(x)]⁰^−jⁿ^(x)[mxjⁿ^(mx)]⁰

m²jn(mx)[xh⁽¹⁾n (x)]⁰−h⁽¹⁾n (x)[mxjn(mx)]⁰

b_n= ^jⁿ^(mx)[xjⁿ^(x)]⁰^−jⁿ^(x)[mxjⁿ^(mx)]⁰

jn(mx)[xh⁽¹⁾n (x)]⁰−h⁽¹⁾n (x)[mxjn(mx)]⁰

(2.14)

x = N_m^2πr_λ , known as the size parameter is of crucial importance since it distinguishes the cluster (x¿1) from the bulk material (xÀ1).

Thanks to the coefficients a_n and b_n, it is possible to determine all the measurable quantities associated with scattering and absorption, such as cross sections. Those cross sections for extinction σ_ext and scattering σ_scat(see Equation (2.15)) are derived by calculating the net rate at which electromagnetic energy crosses the surface of an imaginary sphere centred on the cluster. The absorption cross section σ_abs can be deduced.

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









σ_ext= ^2π_k2

P∞ n=1

(2n+ 1)<{a_n+b_n} σ_sca= ^2π_k2

P∞ n=1

(2n+ 1)(|a_n|²+|b_n|²) σ_abs=σ_ext−σ_sca

(2.15)

In the limit where x ≤ 0.01, meaning that the cluster size is much smaller than the interacting wavelength, the fields inside the cluster are seen as constant. One can make in such a case thequasi-static approxima- tion consisting of taking only the dipolar electric mode (a₁) into account.

The Mie expression of the extinction cross section is simplified consider- ably, as shown in Equation (2.16)^∗.

σ_ext(ω) = 12πω

cε^3/2_m r³ ε₂(ω)

[ε₁(ω) + 2ε_m]²+ε₂(ω)² (2.16) whereε_mandε=ε₁+iε₂are the dielectric functions of the surrounding medium (real) and the cluster (complex), respectively. In the present work, the fitting of the absorption data has been done using this approx- imation, since we consider Ø 10 nm clusters. It is important to note that the expression (2.16) neglects scattering effects, which are likely to be small in the case of clusters as small as 10 nm. Indeed, the scattering becomes a dominant effect when the cluster size is much larger than the electron mean free path (from a diameter of typically 30 nm for Ag⁶⁶).

To make use of this theory, the second part (material) has to come into play. In particular, one has to include in it the size and the dielectric functions of the clusterε(ω, r) and the surrounding medium ε_m.

The Dielectric Function

The dielectric functionε(ω) =ε₁(ω) +iε₂(ω), also called the permittivity oroptical material function, is a complex function describing the intrinsic properties of the material. It determines the position and the shape of the surface plasmon absorption peak. The dielectric function is related to the complex index of refractionN =n+ik by :

½ ε₁(ω) =n²−k²

ε₂(ω) = 2nk (2.17)

∗A simple analytical derivation of Equation (2.16) was performed later on by Genzel and Martin⁶⁵.

Expanding the horizon of molecular electronics via nanoparticle assemblies

E XPANDING THE H ORIZON

VIA

M OLECULAR E LECTRONICS

OF

N ANOPARTICLE A SSEMBLIES

Laetitia Bernard

Laetitia Bernard

!

Expanding the Horizon of Molecular Electronics

via

Nanoparticle Assemblies

Doctor of Philosophy

Table of Contents

Table of Abbreviations

A few Common Symbols

1

A Clever Combination : Organic Molecules &

Nanoparticles

Outline

2

A Short Guide to Organic Molecules & Nanoparticles

2.1 Insights on Molecular Physics

A V

z

y

x

m

m

r

S

2.2 Insights in Cluster Physics

E XPANDING THE H ^ORIZON

M ^OLECULAR E ^LECTRONICS

N ANOPARTICLE A ^SSEMBLIES